use the definition to find the first derivative of f(x)=square rote of 5x+2

Question

use the definition to find the first derivative of f(x)=square rote of 5x+2

Answer 1

Limit as h→0 of {√[5(x + h) + 2] - √(5x + 2)} / h = ?

 

Note the numerator has the form √a - √b.  If you multiply numerator and denominator by √a + √b, you will get

 

(a - b) / [h(√a + √b)].

 

√a = √[5(x + h) + 2]

√b = √(5x + 2)

 

The expression becomes

 

[5(x + h) + 2] - (5x + 2)] / (h{√[5(x + h) + 2] + √(5x + 2)})

 

This simplifies to

 

5h / (h{√[5(x + h) + 2] + √(5x + 2)})

 

Note an h conveniently cancels out and you are left with

 

5 / {√[5(x + h) + 2] + √(5x + 2)})

 

Now let h→0 and you are left with

 

(5/2) / √(5x + 2)

 

And that is what you get for dy/dx when you apply the power rule and chain rule.